Josh Howe

## About Josh Howe

I’m currently a
graduate student working in the San Francisco Bay Area, and I play as much
Magic: the Gathering as I can in my spare time. I started
playing around the release of Mirage, but played only sporadically until the
release of Time Spiral. At the time, I
was doing my undergraduate work at Penn
State and stumbled across
a group of people drafting. I was
immediately hooked again, and in the years since, I’ve made it my goal to
improve as much as a player as I reasonably am able to. I’ve spent the past couple of years trying to
grind my way onto the Pro Tour through PTQs, GPs, rating, or whatever other
means might be available to me, and have had a handful of PTQ top 8s among
other near misses along the way.

## Know Howe – Maximizing EV on MTGO

Initially, I wanted to write about Scars of
Mirrodin sealed deck, given that it’s the PTQ format for the next few
months. Unfortunately, my results thus
far at Scars of Mirrodin sealed (though a very small sample space) have been
almost entirely lackluster. I’ve not had
access to enough sealed deck events to really feel like I have been able to fix
my weaknesses, and so I don’t find myself in a position to advise others on the
new sealed deck format just yet. That is
about to change, though, as Scars of Mirrodin has become available on Magic:
the Gathering Online (MTGO), and as a result the amount of sealed deck to which
I’ll have access will increase enormously.

This increased access should mean I can
play as many sealed deck events as I can reasonably find time for, but there’s
one caveat, and it’s the reason most commonly cited when I ask others why they
don’t play MTGO – cost. As a graduate
student, I have a fairly limited discretionary budget, which means I can’t
actually guarantee that I may play MTGO as much as I am able to, especially if
I also want to be able to play in local tournaments (which are basically a
money sink and don’t have reasonable “freeroll” possibility) and PTQs (let
alone budgeting for GP flights). So when
I play MTGO, I try to maximize the expected value (EV) of my choice in games as
I imagine many players on a budget would like to.

In trying to maximize EV to “break into”
MTGO so that I have an established account there that is profitable enough that
playing requires minimal financial input, I have tried multiple approaches on
multiple occasions, and I’m going to use that experience (along with some data
provided by other friends who have tried similar things) to give some advice to
those looking to get into MTGO, but unsure of how to do it without prohibitive
financial commitment to virtual cards.
When I first started playing MTGO, it was rather daunting, but the
environment quickly became something with which I was much more
comfortable. While this article is more
about how to make choices maximizing EV on MTGO than it is about how to use
MTGO, I’ll first discuss a little bit about the environment and getting started
for those who are not familiar with it.

**Learning
MTGO**

**Setting
up:** The
first step is to download the necessary software and install it. Once the software is installed and updated,
you will have to create an account.
Unfortunately, there’s a cost associated with creating an account ($9.95),
but the good news is that they give you a Magic 2010 booster (this seems like a
strange choice, so I’d hope they’d update this to Magic 2011 at some point) and
2 Magic online Event Tickets (tix) among an assortment of other items
(Planeswalker deck pack, avatars, et cetera).
The booster should be able to resell to dealer bots (more on this later)
for ~3 tix (a pessimistic estimate, as at the time of writing Magic 2011
boosters are selling to bots for 3.36, although I can’t speak for Magic 2010),
so the nominal cost seems to be about $5.

**Learning
the interface:**
This is actually free, and it’s something that is quite worth
doing. With your starter product, you
should be able to venture over to the new players area and get comfortable with
the interface. Go to Menu/Play/Casual
Play/New Players. Here you can join
existing games waiting for players or create your own game, which another
player will join. I strongly suggest
putting a few hours into this and learning how the MTGO interface works; it’s always
frustrating to lose to clicking through combat, but it’s even more frustrating when
it happens in a draft you paid to enter.
Other features worth investigating are the Collection tab (where you can
see everything you own) and the Menu/Community/Marketplace/Classifieds tab
(where trading happens). One final thing
to check out is the Menu/Settings tab, where you can (under Game Play) change
where in the turn it will ask for priority passes. As a note, these can also be changed by
clicking on the appropriate part of the turn mid-game, but it’s good to
establish a good base set of stops that suits your play needs (and so you’re
not surprised when your opponent moves to combat and you don’t have a chance to
use Blinding Mage to keep you from dying… which may have happened to me in one
of my earlier drafts before I learned the interface).

**The
competitive offerings:** MTGO has a variety of competitive offerings,
some of which are “standard fare” and offered all the time (or on a schedule)
and others that are limited-time offerings for prerelease or release of new
product. There are various types of
draft, sealed deck, and constructed events from which a player may choose. Many of these I consider in the next section
in the context of choosing events that maximize your EV as a player.

**Maximizing
EV**

In order to evaluate EV, we need to
consider the probability of winning a match so that we can project likely
performance in a tournament. In the
subsequent analyses, I will use as an example a match win probability
calculated from my performance in sanctioned events in the past year. In order to calculate this match win
probability, I’m counting only matches I’ve won or lost in the past year that
are reported under the “Personal Information Center” portion of the DCI’s
website by looking at the “Ratings History” tab under “Total Rating.”

As it turns out, in the past year, I’ve had
298 matches that have been sanctioned and have had either a win or loss
result. I’ve posted 200 wins, 98 losses,
so my match win probability is just over 0.67; I’ll use 200/298 for my EV
calculations. There are a number of
problems with this method: the pool of
people I’ve played is different from the pool of players on MTGO, this data
considers also (dis)advantages from the human side of gaming that are not so
relevant on MTGO, this assumes I’m as familiar with MTGO as I am with paper
Magic and thus won’t make errors, and only 300 data points may not be enough
for a great estimate of probability.
That said, I think this is the best means I have of calculating my win
probability. The opponents considered
are those who choose to participate in sanctioned events and they come from a
range of events from FNMs and prereleases up to Grand Prix. I could add more data points, but I think
0.67 is probably a pretty fair estimate of a match win probability. If you have another way you’d calculate your
match win probability without a huge MTGO sample space (which would probably be
a better way to do this exercise), please share it in the comments section.

In order to do the calculations, I’m going
to use an Excel spreadsheet to calculate numbers, which I’ll report. I’ll briefly outline the math used in each
case (or reference how to do the calculation) so that if you’re not familiar
with these sorts or probability calculations already, it shouldn’t be too
difficult to reproduce these value or (more relevantly) calculate values to
guide your own choices.

Since the events being considered have
different costs, considering the likely nominal gain or loss (expected outcome
minus entry fee) is less useful than considering the gain or loss scaled by the
entry fee (expected value).

As a note, all analyses neglect the likely
value of any cards received. It is
likely that you can resell some of the cards (or all of them) to bots or other
players for some gain. This is something
to be considered as a small factor in the EV of an event, but it is neglected
here. Additionally, I’m neglecting any
bonus swag (such as prerelease cards) and MOCS qualifier points that can be
awarded.

**Drafts**

There are three primary sorts of 8-man
draft offerings on MTGO: 8-4s, 4-3-2-2s,
and swiss. The first two are single
elimination, with the 8-4s giving 8 packs for 1^{st} and 4 packs for 2^{nd}
place and the 4-3-2-2s comparably giving 4, 3, 2, and 2 packs for 1^{st}
through 4^{th} place respectively.
Swiss drafts award one pack per match win, and each player has the
option to play in up to 3 matches.

To calculate the expected outcome, find the
probability of each outcome. In the case
of single elimination events, we need only consider the probability of winning
some number of rounds (and no more) up to the maximum number of rounds. The total probability must sum to 1, so using
“P” to be the probability of winning, the probability of each outcome is:

0-1:
1-P (0.329)

1-1: P-P^{2 }(0.221)

2-1:
P^{2}-P^{3} (0.148)

3-0:
P^{3} (0.302)

Multiply each outcome by the prize awarded
for that outcome to find the probable outcome of an event. In the case of an 8-4:

0.302*8+0.148*4 = 3.01 packs expected
outcome

For a 4-3-2-2:

0.302*4+0.148*3+0.221*2 = 2.10 packs
expected outcome

To consider a swiss draft, a slightly
different approach is needed, since losing a match does not end the event for
you. Thus, all matches end in either a
win (with probability P) or a loss (with probability 1-P). While there is only one way to go 3-0 or 0-3
(win or lose all matches), there are three ways each to go 2-1 or 1-2 (for 2-1
you can lose the first, second, or third match, and all return the same
prize). A general formula for the
multiplicity factor by which we must weigh each outcome is:

Multiplicity = Rounds!/(Wins!*Losses!),
where “!” denotes a factorial. 4
factorial, for example, = 4*3*2*1 = 24, and 3 factorial = 3*2*1 = 6. As a note, 0! = 1.

This gives the probability of each outcome
as:

0-3:
(1-P)^{3} (0.036)

1-2:
P*(1-P)^{2} (0.218)

2-1:
P^{2}*(1-P) (0.444)

3-0:
P^{3} (0.302)

So the expected outcome is:

0.218*1+0.444*2+0.302*3 = 2.01 packs.

Since all drafts have the same entry fee,
we can directly compare the outcomes and see that the 8-4 gives us the best
value. As a note, when P = 0.5, the
outcome of both swiss and 8-4 is 1.5 packs, while 4-3-2-2 is 1.375. For P < 0.5 swiss gives us the best
expected outcome. For P > 0.5 the 8-4
gives the best expected outcome. For no
value of P does the 4-3-2-2 return the best expected outcome. To compare events with variable entry fee, we
need a normalized way of looking at the returns we are likely to win. So we define an expected value as the
difference between event cost and expected outcome divided by the event cost.

In the case of drafts, the cost is
generally 3 packs + 2 tix (although sometimes drafts are run on a “nix tix” or
“tix only” basis and this will change the cost index… adjust calculations
accordingly. Because we rely on a dealer
economy where packs we have or are selling are worth less than packs we want to
acquire, I value prize packs at 3.5 tix
and packs for entry at 3.8 tix across the board. Therefore, the cost of a draft is 3.8*3+2 =
13.4 tix

This gives us as expected value for each of
the above cases:

**8-4:**
-0.213

**4-3-2-2:**
-0.453

**Swiss:**
-0.474

This value represents the likely return (in
fractional tix) per tix put in. Thus,
for all drafts, we’re likely to lose money over time, but if we keep drafting,
we lose money most slowly to drafting 8-4s.

**Sealed**

Sealed events on MTGO are a bit more scarce
than drafts, and they come in 4-pack and 6-pack variety. The entry fee for 4-pack sealed is 4 boosters
of the appropriate type, while for 6-pack the entry (in the case of prerelease
sealed events) is 30 tix. For 6-pack
release sealed, the entry fee is 24 tix.
These events are typically less populated than drafts, and the offerings
seem to change on a semi (ir)regular basis, so it’s good to keep an eye on
what’s going on in sealed. For the above
three sorts, the expected outcomes and values are provided below:

**4-pack:**
These are 3-round swiss events and provide 5 packs for 3-0, 3 for 2-1,
and 1 for 1-2. Calculations are done
analogously to those for the swiss drafts.

- Expected outcome: 3.06 packs
- Cost:
3.8*4 = 15.2 tix
- EV:
-0.295

**6-pack prerelease:** These are 4-round swiss events that provide
10 packs for 4-0, 4 for 3-1, and 1 for 2-2.
Calculations are again analogous to swiss drafts excepting that the
multiplicity for 4-0 is 1, 3-1 is 4, and 2-2 is 6.

- Expected outcome: 3.91 packs
- Cost:
30 tix
- EV:
-0.544

**6-pack release:** These are 4-round swiss events that provide
13 packs for 4-0, 8 for 3-1, and 3 for 2-2.

- Expected outcome: 6.70 packs
- Cost:
24 tix
- EV:
-0.024

**Constructed**

Constructed events are a bit tricky, as
they require a constructed deck to be owned in order to participate. You need to evaluate how long a deck will
likely be “good” along with how frequently you can play it and weigh its cost
against these factors to decide if it is a good purchase. For these analyses, I’m going to assume that
we have the deck we’d like to play and will post the same win probabilities for
consistency. I don’t have as much faith
in my analyses here as in the limited events due to these extra factors.

Constructed events are offered in “heads
up” style (1 v. 1) and 8-player single-elimination.

**Heads-up:**
These cost 2 tix to enter and award 1 pack to the winner.

- Expected outcome: 0.671 packs
- Cost:
2 tix
- EV:
0.174

**8-player:**
These cost 6 tix and award 5 packs for 1^{st}, 3 for 2^{nd},
and 2 each for 3^{rd} and 4^{th}.

- Expected outcome: 2.4 packs
- Cost:
6 tix
- EV:
0.398

## Additional Events

Other events are offered on MTGO such as
premier events that have variable entries but fixed payouts. The expected values of these types of events
are harder to calculate because they have non-fixed numbers of entrants and
fixed payouts for top performers. They
tend to be more top-heavy (so they reward players who have higher win
probabilities), but the EV is based on the number of entrants and how many of
them drop before they are out of prize contention. For these, my recommendation is that you try
to participate in ones on off-peak hours, as the entry numbers in these tend to
be lower so your EV is requisitely higher since fewer people are sharing more
payout.

One final event type that is a fixed-entry
event is a release event draft, the “64-player ‘Top 8’ Premier draft.” These drafts have a fixed entry of 64 players
and are played in eight 8-player pods that cut a winner from each pod to
compete in a “Top 8” draft. The events
are single elimination with payouts of 20 packs for 1^{st}, 15 for 2^{nd},
10 for 3^{rd}-4^{th}, 6 for 5^{th}-8^{th}, 4
for 9^{th}-16^{th}, and 2 for 17^{th}-32^{nd}. Cost to enter is 4 tix + 3 packs. Calculations are done as if for a 6-round
single-elimination draft.

- Expected outcome: 4.80 packs
- Cost:
15.4 tix
- EV:
0.090

**Conclusions**

Overall, the constructed events return the highest
EV. This is not unexpected, because the
overhead cost of a deck is not factored into the event entry costs. In fact, it makes sense that if you play with
a constructed deck (that is viable) enough, you can make the money back that
you spent on the deck. The problem with
constructed on MTGO as a source of maximizing EV is that it requires a lot of
time and play to make back the cost of the deck. If you’ve got a lot of time to play and the
equity to put into a deck up front, then this is likely the best way to go
about it. You can buy cards from and
resell cards to bots without major losses, so you can have a fairly dynamic
deck without too much expectation of financial loss, and a lot of play in 8-man
queues should be a good way to profit from your investment.

In limited events, two events stand out
from the rest: release event 64-player
drafts and release event 16-player sealeds.
Both of these events happen only for a two-week window around the launch
of a set. Scars of Mirrodin release
events run October 20^{th} through November 3^{rd}, so now is a
particularly convenient time to be doing limited online. While the EV on the sealed event with highest
EV is still negative by about 2 cents per dollar (in the case of the winning
probability used here), the value of cards opened should more than compensate
for that minor loss of value.

In the case of M11, I played the release
event sealed deck tournaments almost exclusively during the lead-up to Grand
Prix: Portland for practice. I put about $40 into MTGO and used it to fund
my play and have yet to redeposit. I am
left with a nearly complete M11 set (which I can easily complete to “cash out”
as a physical set if I’d like in order to recoup the financial investment),
which I can also use as a reasonable basis for starting a constructed
deck. I plan to use the same approach to
practice for PTQs with Scars of Mirrodin release events.

If I had to prescribe a “maximum EV”
approach to MTGO, it would be the following:

During set release event windows, “grind”
64-player drafts and 16-player sealed queues.
Use these to build sets to either build constructed decks or cash out,
depending on needs. Early on in the set,
resell cards to bots frequently (as values tend to start inflated due to need
for players looking to play constructed immediately), but curtail this once
prices begin to relax to complete sets if that’s something of interest to
you. Generally, completing sets and
cashing them out and reselling the paper copies is more lucrative than selling
the singles off to bots, so if you can make this work, it’s your best option.

After release events, use the equity gained
in your collection through grinding to put together a constructed deck, and use
this to grind the 8-player constructed queues.

While these numbers are not valid for
everyone, I encourage you to do these calculations with your results if you
want “more accurate” numbers. In
general, the results I found for myself should not be atypical. In fact, most of you will likely find the
same outcomes – the limited events that I prescribe as “highest EV” are so much
more lucrative in prize structure that the dependence of relative EV between
these and other events on win probability actually weakens. That said, one case where knowing your win
probability can really help is if you want to practice drafting. All else aside, there can be a case for doing
swiss (or 4-3-2-2) drafts rather than 8-4s if you just need bulk practice for
minimal costs. Granted, the mean player
skill level changes between these drafts and it may change both your win
percentages and the relevance of your practice, so bear this in mind.

Hopefully you have found this to be
insightful and have found that it has provided a reasonable case for these
sorts of analyses in terms of the value of various tournament offerings. I personally find these analyses to be useful
when determining whether or not to participate in events. Granted, there are always intangibles for
events in person (and even online) such as gain to play skill through practice. These things have value. A Grand Prix tournament will seldom have a
positive EV unless you happen to live in an area of the world with small GPs
and one shows up in your back yard – I still advocate traveling to and
attending these because they are just that much fun. But for online events, where the enjoyment is
more marginal and the practice value is comparable between variable offerings,
the financial EV is a good test for how worthwhile an event likely is. While my time to play MTGO is quite limited
right now due to academic commitments, I will be grinding Scars sealed deck as
much as possible during the two weeks of high EV release events. I hope to see some of you in the queues! As always, please feel free to leave
comments, in this case especially if you think I have made an error or
overlooked something major.

Josh Howe

Maniacal42 on MTGO

## Comments

Good read, appreciate the formulas.

I love mtgo, though i get the most hideous runs of bad luck it can be quite humorous, draw 6 land, mull draw 1, mull draw 5 land etc.

But i suppose ya play enough games itll happen, and its only the OMG I CANT BELIEVE IT things that tend to stick out in your mind ....

Although at the end you mention reselling the cards you played with you didnt include them in your calculations. Isnt there an average price you can put on an opened booster? If that price is say 2 tix then that is an additional 12 tix coming out of a sealed event which would have a large impact on your EVs.

67% winnings? I think that is way toooo much optimistic! IRL maybe but online?

Cards sell for pennies on mtgo but even if you could get 1 ticket back per booster opened it may be enough to swing some of these EV's positive. And there will be occasions were you open a primeval titan or something which would have a huge impact on the calculations. This is contingent on selling everything of course.

The daily events have a standardized payout now (all based on 4 rounds of swiss, not attendance) and could probably use a look.

Here's my EV info :

Never, ever play 4322s unless they are nix tix. Play a lot when there is a format you like/do well at/has a chance to open a JtMS.

It's worked out great for me (except for the open JtMS bit). I was very successful in RGD, Coldsnap, Time Spiral (just the first set), Alara (not the block, just Shards), and Scars sealed (not draft ... yet). I've been good with the base sets too, but with half reprints...meh.

Also, for you fellow old timers who remember playing, say, MVW limited, the old sets have decent resale value for their chase cards. (MVW's 'goodbye' events is where I gravied my going infinite - winning an entire set of a pretty valuable block still has me in tix.) Also, leagues were awesome EV for an experienced player (I started my grasp on 'infinite' in Coldsnap leagues). Too bad WoTC keeps lying about their return.

And don't play (unless you just have that itch) in formats where you really lose a lot or just plain don't like (the full TPF crushed me, as well as Shards block - I played a ton of SSC for Jace but no luck; and I loathed Lorwyn limited)

GL to any of you budget watchers to try and maximize both your fun and keep it easy on your wallet.

Thanks for the feedback; to answer questions:

Expected return from an opened booster is certainly less than 2 tix, probably on the order of 0.5-1 tix, and that's including the odds of ripping a pretty sick mythic. If I get a chance later, I'll run some numbers and put up a ballparked estimate on pack EV based on projected buy prices. Dealers are selling Koth at 28, Opal at 12.75, Venser at 14, Elspeth and 12.75, Masticore at 13... if I recall correctly. Everything else is basically junk (Ratchet Bomb they're buying for 3, duals at 0.5, Platinum Emperion at 0.5, based on what I saw yesterday). I'm not sure how this correlates with buy prices for the high-end mythics since my opens have been pretty savagely awful so far (granted I'm only 3 sealeds deep in practice). As it stands, the average return from a sealed is likely about 3-4 tix. Certainly not insignificant, but it's hard to generate a good comparison of this against draft where you have to spend picks on grabbing the things you'll later sell... which in turn influences your win probability... as you can see, it becomes very tough to get good numbers for this in terms of draft. I could modify the sealed deck EVs, but that would be unfair to draft... so I omitted these values.

I feel like 0.67 is a pretty high win probability, definitely. I was surprised at my performance being this strong over the past year; I'd have expected about 0.62. And you'd be surprised, but as much as people say that MTGO is tougher than paper magic, I've not really found that to be the case. Granted, I play in a fairly competitive geographic area, so a lot of my regular paper opponents also play MTGO. Aside from this, there is the concern about "is this a valid way to compute match win probability?" I've had a friend of mine raise the concern that there is an associated bias with selectively removing myself from tournaments in the event of poor performance (such as an 0-2 drop), while a good performance means I continue to play, thus inflating P. To be fair, I had not considered this, and am working through a calculation on a sample space of 1000 PTQs to see if I can back out a correction factor to P. I'm busy today, so I'm not sure if I'll get a chance to do it, but I'll cross-post the math to the comments here if I do get to do this.

I was unaware of DEs having standard payout and entry now, so I can definitely take care of running numbers on those (or you should be able to if they're based on 4-round swiss). Multiplicity should look like 4!/(4!0!) 1, 4!/(3!1!) = 4, 4!/(2!2!) = 6 (these all just binomial coefficients).

Good article Josh. I do these calculations myself and I use a 50% WR just to keep it simple and generate realistic calculations to account for bad variance. One thing I would add, of which other people take note, is the value of QPs during MTGO championship seasons. If you carry them over into the value of the events that they qualify for, they are worth about 1 ticket each for as many as there are in the prize pool. This makes MTGO much more affordable/profitable during those seasons which begin again in January.

Absolutely fantastic article. Well done.